Imo, It is about setting priorities. Ofcourse the intensity of the League, having nearly every week 2 matches and not having a break during the winter is very hard for the players, but this has been for ages.
A huge difference with the top teams of the EPL and the 3 top teams, Madrid, Barca and Bayern, is that they have a deeper quality bench. They can actually rest their Robben or Ribery during competition games, because players like Muller or Götze take over their spot without any problem.
So the problem is, where does the problem lie? I think it is partly because of the way premier league clubs are buying their players. Ofcourse you have the famous home-grown quotum which puts a premium on talent, but the players who are being brought in the last few years do not cover the gaps by the "deadwood" who are sold. It is like you are willing to make one or two expensive transfers and willing to compensate by letting 5 squad players go. That's one reason why your quality depth isn't as good as the other teams.
Another reason is that the managers don't prioritise the European games as much as other countries do. At this moment it looks like all the managers really care about is playing in the 1/8th finals against the top top teams. They forget to have a chance to play against them, you first have to reach to the 1/8th finals, and after that you need to have enough experience how to beat european teams. To me it seems like the managers rate a game against mid-table teams in the BPL higher than a match in the group fase against a 2-4th seeded club.
TL;DR Invest in quality depth and prioritise the european games over BPL games like all other clubs are doing.